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initial and boundary value problem of fuzzy fractional-order nonlinear Volterra integro-differential equations

JOURNAL OF APPLIED MATHEMATICS AND COMPUTING 2023年第2期69卷 1765-1793页

作者： Agilan, K. Parthiban, V Vellore Inst Technol Sch Adv Sci Chennai Campus Chennai 600127 Tamil Nadu India

The fractional derivative in Caputo sense for the class of fuzzy fractional order Volterra integro-differential equations of the first kind is examined in this article. This paper considers both the initial and boundary value problems simultaneously. The transformation of first kind to second kind is done using Leibniz rule. Fixed point theory is used to establish the existence and uniqueness of the considered equation in its second kind. Furthermore, the Adomian decomposition method is used to determine the solution to the proposed problem. We provide some examples to back up the approach. The numerical and graphical representations of the symmetry between lower and upper cut representations of the fuzzy solutions are shown using MATLAB. A graphical representation is supported to critically examine how such methods work.

关键词： Fuzzy fractional differential equations Leibniz rule Fixed point theorem initial and boundary value problem Existence and uniqueness Adomian decomposition method

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Existence and uniqueness of solutions to initial boundary value problem for the higher order Boussinesq equation

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NONLINEAR ANALYSIS-REAL WORLD APPLICATIONS 2019年 47卷 436-445页

作者： Oruc, Goksu Muslu, Gulcin M. Istanbul Tech Univ Dept Math TR-34469 Istanbul Turkey

This paper studies the initial and boundary value problem for the higher order Boussinesq equation. The existence of a local and global solution are proved. The sufficient conditions for finite time blow-up solutions ... 详细信息

关键词： The higher order Boussinesq equation initial and boundary value problem Existence of local solution Existence of global solution Blow-up Concavity method

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A new neural network method for solving Bratu type equations with rational polynomials

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INTERNATIONAL JOURNAL OF MACHINE LEARNING AND CYBERNETICS 2025年第2期16卷 1355-1369页

作者： He, Jilong Cao, Cong Huzhou Univ Dept Math Huzhou 313000 Zhejiang Peoples R China Changsha Univ Sch Math Changsha 410022 Hunan Peoples R China

The Bratu-type equation is a fundamental differential equation with numerous applications in engineering fields, such as radiative heat transfer, thermal reaction, and nanotechnology. This paper introduces a novel approach known as the rational polynomial neural network. In this approach, rational orthogonal polynomials are utilized within the neural network's hidden layer. To solve the equation, the initial boundary value conditions of both the differential equation and the rational polynomial neural network are integrated into the construction of the numerical solution. This construction transforms the Bratu-type equation into a set of nonlinear equations, which are subsequently solved using an appropriate optimization technique. Finally, three sets of numerical examples are presented to validate the efficacy and versatility of the proposed rational orthogonal neural network method, with comparisons made across different hyperparameters. Furthermore, the experimental results are juxtaposed against traditional methods such as the Adomian decomposition method, genetic algorithm, Laplace transform method, spectral method, and multilayer perceptron, our method exhibits consistently optimal performance.

关键词： Neural network Rational orthogonal polynomial Chebyshev polynomials Bratu-type equations initial and boundary value problem

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Importance of considering unsaturated triaxial tests including ceramic disk as initial and boundary value problems

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SOILS AND FOUNDATIONS 2021年第3期61卷 901-913页

作者： Yoshikawa, Takahiro Noda, Toshihiro Kodaka, Takeshi Nagoya Univ Dept Civil Engn Chikusa Ku Furo Cho Nagoya Aichi 4648603 Japan Nagoya Univ Disaster Mitigat Res Ctr Chikusa Ku Furo Cho Nagoya Aichi 4648603 Japan Meijo Univ Dept Civil Engn Tempaku Ku 1-501 Shiogamaguchi Nagoya Aichi 4688502 Japan

Considering unsaturated triaxial tests including a ceramic disk as initial and boundary value problems, a series of suction change, isotropic consolidation, and drained shearing processes was simulated using a soil-water-air coupled elastoplastic finite deformation analysis code. As a result, it was demonstrated that the delayed behavior of a wetting-induced collapse during the suction change and isotropic consolidation processes, as well as water-absorption behavior during the subsequent drained shearing process, were attributed not to the characteristics of the soil specimen, but to the permeability of the disk. Thus, it is important to take the ceramic disk into account in unsaturated triaxial tests regarded as initial and boundary value problems in order to understand the temporal change behavior of unsaturated soil specimens. Otherwise, there is a risk that the behavior appearing as the solution of an initial and boundary value problem may be modeled as a constitutive relation. (C) 2021 Production and hosting by Elsevier B.V. on behalf of The Japanese Geotechnical Society.

关键词： Unsaturated soil Soil-water-air coupled finite deformation analysis Elastoplastic constitutive model initial and boundary value problem Triaxial test Ceramic disk

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An accurate and cost-efficient numerical approach to analyze the initial and boundary value problems of fractional multi-order

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COMPUTATIONAL & APPLIED MATHEMATICS 2018年第5期37卷 6582-6600页

作者： Sayevand, K. Tenreiro Machado, J. Malayer Univ Fac Math Sci POB 16846-13114 Malayer Iran Inst Engn Polytech Dept Elect Engn Porto Portugal

This paper proposes an cost-efficient method for the numerical solution of fractional differential equations of multi-order. The suggested algorithm adopts the operational matrix of fractional derivative in term of Caputo sense using modified cubic B-spline functions. The new method leads to a system of algebraic equations that improves the solution accuracy and reduces the computational complexity. Several numerical examples illustrate the applicability of the proposed method.

关键词： Caputo fractional derivative Cubic B-spline initial and boundary value problem Operational matrix 34A08 41A15 34K10 34E20

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ANALYSIS OF CHEBYSHEV PSEUDOSPECTRAL METHOD FOR MULTI-DIMENSIONAL GENERALIZED SRLW EQUATIONS

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Applied Mathematics and Mechanics(English Edition) 2003年第10期24卷 1168-1183页

作者：尚亚东郭柏灵 Department of Mathematics Guangzhou University Institute of Applied Physics and Computational Mathematics

The Chebyshev pseudospectral approximation of the homogeneous initial boundary value problem for a class of multi-dimensional generalized symmetric regularized long wave (SRLW) equations is considered. The fully discrete Chebyshev pseudospectral scheme is constructed. The convergence of the approximation solution and the optimum error of approximation solution are obtained.

关键词： multi-dimensional generalized SRLW equation initial and boundary value problem Chebyshev pseudospectral method error estimate

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Global existence for a singular phase field system related to a sliding mode control problem

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NONLINEAR ANALYSIS-REAL WORLD APPLICATIONS 2018年 41卷 128-151页

作者： Colli, Pierluigi Colturato, Michele Univ Pavia Dipartimento Matemat Via Ferrata 5 I-27100 Pavia Italy

In the present contribution we consider a singular phase field system located in a smooth and bounded three-dimensional domain. The entropy balance equation is perturbed by a logarithmic nonlinearity and by the presence of an additional term involving a possibly nonlocal maximal monotone operator and arising from a class of sliding mode control problems. The second equation of the system accounts for the phase dynamics, and it is deduced from a balance law for the microscopic forces that are responsible for the phase transition process. The resulting system is highly nonlinear;the main difficulties lie in the contemporary presence of two nonlinearities, one of which under time derivative, in the entropy balance equation. Consequently, we are able to prove only the existence of solutions. To this aim, we will introduce a backward finite differences scheme and argue on this by proving uniform estimates and passing to the limit on the time step. (C) 2017 Elsevier Ltd. All rights reserved.

关键词： Phase field system Maximal monotone nonlinearities Nonlocal terms initial and boundary value problem Existence of solutions

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The well-posedness of the double-diffusive convection system in a bounded domain

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MATHEMATICAL METHODS IN THE APPLIED SCIENCES 2018年第11期41卷 4327-4336页

作者： Chen, Fei Guo, Boling Zeng, Lan Inst Appl Phys & Computat Math Beijing 100088 Peoples R China China Acad Engn Phys Grad Sch Beijing 100088 Peoples R China

In this paper, we consider the well-posedness of the initial and boundary value problem for the double-diffusive convection system in a bounded domain Omega subset of R-3. We first prove the local existence of the unique strong solution with large initial data in H-0(1) (Omega) and then extend it to a global one provided the L-2 norm of the initial data is small.

关键词： double-diffusive convection initial and boundary value problem well-posedness

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Mathematical Analysis of a Seawater Intrusion Model Including Storativity

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SIAM JOURNAL ON MATHEMATICAL ANALYSIS 2017年第1期49卷 29-63页

作者： Choquet, C. Diedhiou, M. M. Rosier, C. Univ La Rochelle Lab MIA EA 3165 F-17000 La Rochelle France Univ Lille Nord France CNRS FR 2956 LMPA J Liouville ULCO F-62228 Calais France

This paper is devoted to a seawater intrusion model in free aquifers, including rock and fluid compressibility effects. This original model is based on the superimposition of a phase-field model on sharp front tracking. It leads to the analysis of a two-dimensional strongly coupled system of PDEs of parabolic type describing the evolution of the depths of the two free surfaces. The main difficulty is dealing with the strong nonlinearity occurring in the time derivative due to the fluid storage coefficient. This point yields to the manipulation of very weak solutions solutions of an alternative variational formulation focusing on the structural weakness of the problem.

关键词： seawater intrusion problem nonlinear parabolic PDEs system of strongly coupled PDEs initial and boundary value problem

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Triaxial test on water absorption compression of unsaturated soil and its soil-water-air-coupled elastoplastic finite deformation analysis

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SOILS AND FOUNDATIONS 2020年第5期60卷 1151-1170页

作者： Yoshikawa, Takahiro Noda, Toshihiro Nagoya Univ Dept Civil Engn Chikusa Ku Nagoya Aichi 4648603 Japan Nagoya Univ Disaster Mitigat Res Ctr Chikusa Ku Nagoya Aichi 4648603 Japan

Triaxial tests and numerical simulations were conducted to determine the water absorption compression of unsaturated soil. In the triaxial tests, unsaturated silt specimens under the same initial condition were prepared and the processes of different suction reduction, isotropic consolidation, and exhausted-drained shearing were applied to the specimens. The numerical simulations used a soil-water-air coupled finite deformation analysis code with a newly incorporated elastoplastic constitutive model. The degree of saturation was introduced to the internal state variables of the yield function of the SYS Cam-clay model and a soil-water characteristic model with void ratio dependency. Then, the triaxial test was treated as an initial and boundary value problem, and the series of processes in the experiments were simulated using a single set of material constants and initial values. The new findings are as follows: 1) The loading criterion and the classification of loading conditions considering change in saturation degree can be specified in the elastoplastic constitutive model for unsaturated soil. 2) Water absorption compression behavior was observed in the experiments not only under suction reduction but also under constant suction condition. This water absorption compression under constant suction condition was explained as a saturation-increase behavior due to volume compression using the soil-water characteristic model with void ratio dependency. 3) By considering the triaxial test as an initial and boundary value problem, the temporal change in the volume and the amount of water absorption/drainage were able to be reproduced. (C) 2020 Production and hosting by Elsevier B.V. on behalf of The Japanese Geotechnical Society.

关键词： Unsaturated soil Soil-water-air coupled finite deformation analysis Elastoplastic constitutive model Soil-water characteristic model initial and boundary value problem Triaxial test

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